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## Life Tables

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**Life Tables**STA 220 – Lecture #7**Life Tables**• A life table is a table which indicates the probability of someone (or something) • Used extensively in the insurance business • Life Insurance • Based on gender, smoking status, current age, etc.**Life Tables**• Refer to previous slide • P(male alive at 60) = 0.755 = 75.5% • P(Female alive at 60) = 0.863 = 86.3% • Males are not expected to live as long as females**Life Tables**• Life tables can be used to predict the following values: • What’s the probability of being alive at a certain age? • What’s the probability of not being alive at a certain age? • What’s the probability of dying (also known as risk) between two ages? • Given that a certain age has been reached, what’s the probability of living to a specific older age? • What’s the life expectancy? • Given that a certain age has been reached, what’s the life expectancy?**P(alive at a certain age)**• To determine the probability of being alive at a certain age, go to the life table and read off the probability of being alive. • P(male alive at 25) = • P(female alive at 50) =**P(alive at a certain age)**• P(male alive at 80) = • P(female alive at 95) = • P(male alive at 105) = • P(female alive at 106) =**P(not alive at a certain age)**• People or things have 2 basic states: alive (or functioning) and not alive (or not functioning) • Sum of these 2 states must equal 1 • P(alive at some age) + P(not alive at some age) = 1 • P(not alive at some age) =**P(not alive at a certain age)**• Remember the life table contains the value for P(alive at some age) • Suppose we want to find the probability that a male is not alive at age 20 • P(male alive at 20) = • P(male not alive at 20) = 1 – P(male alive at 20) = = 0.041 = 4.1%**P(not alive at a certain age)**• Probability a female is not alive at 35 • P(female alive at 35) = • P(female not alive at 35) = 1- P(female alive at 35) = = 0.040 =**P(not alive at a certain age)**• Probability a female is not alive at 65 • P(female alive at 65) = • P(female not alive at 65) = 1- P(female alive at 65) = = = 19.3%**Risk**• What’s the risk between this age and that age? • What’s the probability of dying between this age and that age? • P(dying between a younger age and an older age) =**Risk**• Suppose you want to find the probability of a male dying between the ages of 20 and 30 • P(male dying between 20 and 30) = P(alive at 20) – P(alive at 30) • P(male alive at 20) = • P(male alive at 30) = • P(male dying between 20 and 30) = = 0.015 = 1.5%**Risk**• Suppose you want to find the probability of a female dying between the ages of 20 and 40 • P(female dying between 20 and 40) = P(alive at 20) – P(alive at 40) • P(female alive at 20) = • P(female alive at 40) = • P(female dying between 20 and 40) = = 0.018 = 1.8%**Risk**• Suppose you want to find the probability of a female dying between the ages of 85 and 90 • P(female dying between 85 and 90) = • P(female alive at 85) = • P(female alive at 90) = • P(female dying between 85 and 90) = 0.262 – 0.130 =**P(alive at an older age given alive at a younger age)**• Life tables reflect the values for a newborn child or a brand new thing • Once a person ages, their probability of reaching an older age increases as the size of the population decreases • P(alive at an older age given alive at a younger age)**P(alive at an older age given alive at a younger age)**• What’s the probability that a female will be alive at 90 given that she lived to 85? • P(alive at 90 given alive 85) • P(alive at 90) = 0.130 • P(alive at 85) = 0.262**P(alive at an older age given alive at a younger age)**• Suppose your best friend’s aunt is 45. What’s the probability that she will be alive to see your best friend graduate from medical school in 10 years? • P(alive at 55 given alive at 45) • P(alive at 55)= , P(alive at 45)=**P(alive at an older age given alive at a younger age)**• What’s the probability that a male will not be alive at 60 given that he is alive at 45? • Recall: P(not alive at some age) = • So, P(not alive at older age given alive at younger age) = 1 – P(alive at older age given younger age)**P(alive at an older age given alive at a younger age)**• P(alive at 60 given alive at 45) • P(alive at 60)=0.755, P(alive at 45)=0.905 • P(not alive at 60 given alive at 45) = 1- P(alive at 60 given alive at 45) =**Calculation of Life Expectancy**• Suppose you wanted to determine the life expectancy of New Englanders in 1900. To do this, you went to various town halls and obtained death records for 1,000 individuals.**Calculation of Life Expectancy**• To calculate a weighted average for the age at death: • Risk*Age + Risk*Age + Risk*Age+…+ • Where risk for a 0 year old is 82/1,000 and risk for a 1 year old is 19/1,000, etc**Calculation of Life Expectancy**• Risk column • Subtract probability of being alive at the current age from the previous age • P(dying between a younger age and an older age) = • Risk between 0 and 10 • P(dying between 0 and 10) = • Risk between 50 and 70 • P(dying between 50 and 70) = 0.60-0.20 =**Calculation of Life Expectancy**• Age1 column • Average of the previous age and the current age • Average age between 0 and 10 = • Average age between 30 and 50 =**Calculation of Life Expectancy**• Risk * Age1 column • Product of the Risk and Age1 • 0.10 x 20 = 2.0 years • 0.20 x 40 = 8.0 years • 0.40 x 60 = • 0.20 x 80 = 16.0 years**Calculation of Life Expectancy**• Life expectancy • Add up the values in the last column • 0.5+2.0+8.0+24.0+16.0 =**Procedure for Computing Life Expectancy**• 5 steps to computing the life expectancy given a life table: • Create a table with column headings of Age, P(alive), , Age1, and *Age1 • Compute risk for each consecutive pair of ages: P(alive at previous age) – P(alive at current age) • Compute the average age for each consecutive pair of ages: Age1=(Previous age+Current age)/2 • Add up all the values in the last column**Calculation of Conditional Life Expectancy**• We already found that if you’re already 20, then the probability of being alive at 30 is higher than the value in the life table • Procedure is same as computing life expectancy with one additional step • Must compute all of the to use in the life table**Calculation of Conditional Life Expectancy**• Suppose we had the same life tables as in the previous example and we wanted to determine the life expectancy for a person that was 30 years old. • Compute following conditional probabilities first • P(alive at 50 given alive at 30) • P(alive at 90 given alive at 30)**Calculation of Conditional Life Expectancy**• P(alive at 30 given alive at 30) = P(alive at 30)/P(alive at 30) = • P(alive at 50 given alive at 30) = P(alive at 50)/P(alive at 30) = • P(alive at 70 given alive at 30) = P(alive at 70)/P(alive at 30) = • P(alive at 90 given alive at 30) = P(alive at 90)/P(alive at 30) =**Calculation of Conditional Life Expectancy**• These values can be substituted into the table for • Need to recalculate column using these conditional probabilities • The column stays the same because the ages have not changed • Last column multiplies the new column with the column • Finally, add up all the values in the column to arrive at the conditional life expectancy**Life Expectancy**• Useful in determining payouts of Social Security to retirees • Should women get the same amount per month as men? • Average life expectancy can be used as a measure of quality or reliability